# Tagged Questions

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Let $A,B\in\mathbb{M}_{n\times n}(\mathbb{R})$ and $A,B$ are symmetric matrics. Prove that if $\vec{x}^TA\vec{x} = \vec{x}^TB\vec{x}$ $\forall\vec{x}$, then $A=B$. Since $A,B$ are symmetric, they are ...
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### When does $x^TAx + c^Tx$ have a global minimum?

This question is closely related to my last question about extended quadratic forms. I figured out a nice criterion, when $$f : \mathbb R^n \rightarrow \mathbb R$$ $$f(x) = x^TAx + c^Tx$$ has a ...
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### Quadratic Form - New Axes = Eigenvectors of P, Order of Eigenvectors Important? [Kolman P539 Example 6]

Hypothesise that $P$ is the symmetric matrix of some quadratic form $g(\mathbf{ x} ) = \mathbf{ x^TAx}$. Then $P$ is the orthogonal matrix consisting of orthogonal eigenvectors of $A$. Moreover, use ...
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### A is a matrix of positive defined quadratic form. How can I show, $A^{-1}$ is the same?

Let a square matrix A is a matrix of positive defined quadratic form. How can I show, that $A^{-1}$ also a matrix of a positive defined quadratic form? Positive defined quadratic form is A(x,y), that ...
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### How to prove: a quadratic form with a matrix $B = CC^T$ is positive defined?

Let a matrix $C \in \Bbb K^{n \mathtt x n} : det(c) \ne 0$ (K is any field - C or R) $\Rightarrow$ a quadratic form with a matrix $B = CC^T$ is positive defined one. How to prove it?
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I know that the quadratic form $x'Ax$ is a concave in vector $x$ if matrix $A$ is negative semi definite. What happens if $A$ depends on $x$ (so that I have $x'A(x)x$), but I still know that $A(x)$ is ...
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### Relaxed matrix factorization

I have an optimization problem like $min~~ \frac{1}{2}||L||^2_F + \frac{1}{2}||R||^2_F,~~~~subject~to~ M=LR^T$ with respect to $L$ and $R$. I know that there are several factorizations that give ...
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### From the quadtratic form of a matrix to its symmetric matrix

I am trying to solve this quadratic form: $$f(x_1 , x_2 , x_3) = x_1^2 + x_2^2 + 5x_3^2 -2x_1x_2 + 6x_1x_2+ 3x_2x_3$$ I know that the quadratic form is defined as: $$f(x)= x^t Qx$$ However my ...
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### Squaring is not monotone [duplicate]

I know that the squaring of operators is not a monotone operation but I can't find a example showing exactly that. I'm looking for (preferably rather simple 2 by 2 matrices) $0<A\leq B$ such that ...
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### bounding operator norm by quadratic forms on orthonormal basis

Let $\{u_i\}_{i=1}^n$ be a set of orthonormal basis and $M$ a symmetric matrix. Suppose $$\left|\langle u_iu_i^T,M\rangle\right|\leq \tau\ \ \forall i=1,\dots,n.$$ Can I ...
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$A$ is an $m\times m$ real matrix in $\mathbb{R}^{m\times m}$. If $x^TAx=0\ \forall\ x\in\mathbb{R}^m$, can we conclude that $A=0_{m\times m}$? Why? Note: $x^T$ is the transpose of $x$. $0_{m\times ... 1answer 63 views ### Extremum of a multidimensional quadratic function I have the following function: $$g(h) = h'\Sigma\Sigma'h-h'm-r,$$ where$h$is a vector in$\mathbb{R}^M$,$\Sigma$is a$M\times K$matrix such that$\Sigma\Sigma'$is positive definite and has ... 0answers 152 views ### Proving the max of a quadratic form${\mathbf x}^T\mathbf A \mathbf x$can be attained when$x$is from$n$-dimensional hypercube updated: Maybe my original question is somewhat misleading. I rewrite some of the post. This is some research problem I'm working on. I have an$n\times n$symmetric positive-definite matrix ... 0answers 49 views ### low rank decomposition of composite PSD matrix The matrix$M=AVA^T - BCB^T +D$is known to be positive semidefinite (PSD), where$V, C, D$are each diagonal matrices with positive values, and$V, C$has small size when compared to the size of$M$. ... 1answer 104 views ### Conditions for$v \otimes v$to be positive semidefinite for complex$v$I have a complex-symmetric matrix (in the sense$A=A^{T}$not$A=A^{H}$), which is required to be positive semi-definite in the following sense (sometimes referred to as positive real):$ \Re(x^{*} A ...
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I have this quadratic form $Q= x^2 + 4y^2 + 9z^2 + 4xy + 6xz+ 12yz$ And they ask me: for which values of $x,y$ and $z$ is $Q=0$? and I have to diagonalize also the quadratic form. I calculated ...
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### How to show that $A=B-C$

How to show that for a real symmetric matrix $A,~A$ can be written as $A=B-C$ where $B,C$ are positive definite real symmetric matrices? Please help me ! I'm clueless.
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### Solving quadratic form $\mathbf{x}^\mathrm{T}\mathbf{A}\mathbf{x} = c$ for $\mathbf{x}$

This is a simple question I hope, is there an easy way to solve: $$\mathbf{x}^\mathrm{T}\mathbf{A}\mathbf{x} = c$$ for $\mathbf{x}$? (Assume $\mathbf{A}$ is positive definite). Geometrically the ...
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### How to prove that $E:=ABC D$ is also positive definite?

Now I think this is true: Let $A$, $B$, $C$ and $D$ be symmetric, positive definite matrices and suppose that $E:=ABCD$ is symmetric. How might I prove that $E$ is also positive definite? ...
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### two non-degenerate quadratic forms on $GF(2)^2r$

I know this: There are two non-degenerate quadratic forms on $GF(2)^2r$. The hyperbolic form may be taken to be $Q^+(x)=x_0 x_1 + \cdots +x_{2r-2}x_{2r-1}$ , and the elliptic form to be ...
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### Matrix Equation with Quadratic form

I am working in a problem that involves multivariate normal distributions and, at a given point, I need to solve the following matrix equation: $$x=\sqrt{x^{\prime}\Sigma^{-1}x} \cdot y$$ Where $x$ ...
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For the Quadratic Form $X^TAX; X\in\mathbb{R}^n, A\in\mathbb{R}^{n \times n}$ (which simplifies to $\Sigma_{i=0}^n\Sigma_{j=0}^nA_{ij}x_ix_j$), I tried to take the derivative wrt. X ($\Delta_X X^TAX$) ...
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### Proving that a matrix is negative definite using its principal minors

I am interested to find out the proof for the following statement (it's from my textbook and it is stated without proof): A symmetric matrix is negative definite if and only if all of its ...